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Reverse Query Processing Carsten Binnig, Donald Kossmann and Eric Lo ICDE 2007

Reverse Query Processing Carsten Binnig, Donald Kossmann and Eric Lo ICDE 2007. Presented by Ankit Shah Bikash Chandra. Motivation. Testing database applications requires generating test databases Test Database is required for

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Reverse Query Processing Carsten Binnig, Donald Kossmann and Eric Lo ICDE 2007

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  1. Reverse Query ProcessingCarsten Binnig, Donald Kossmann and Eric LoICDE 2007 Presented by Ankit Shah Bikash Chandra

  2. Motivation Testing database applications requires generating test databases Test Database is required for • carrying out functional tests on the new application logic • testing the performance of a RDBMS for any user defined benchmark queries • debugging SQL queries

  3. Motivation A number of commercial tools are available that automatically generate test databases. However, • the databases generated by these tools are not adequate for testing a database application. • If an application query is executed against such a synthetic database, then the result of that application query is likely to be empty or contain weird results

  4. Motivation Sample Query SELECT orderdate, SUM(price*(1-discount)) FROM Lineitem, Orders WHERE l_oid=oid GROUP BY orderdate HAVING AVG(price*(1-discount))<=100 AND SUM(price*(1-discount))>=150; Test Database generated by commercial tool

  5. Motivation • Some tools allow the user to specify constraints for generating test databases (e.g., domain ), those constraints are defined on the base tables only. So, the query results can’t be controlled directly. • Therefore, those tools can hardly deal with the complexity of SQL and application programs.

  6. Reverse Query Processing (RQP) • Given a Query Q and a Table R, the goal is to find a Database D (a set of tables) such that Q(D) = R. • Based on a reverse relational algebra (RRA). • Each operator of relation algebra has a corresponding operator in RRA • Unlike traditional query processing, iterators in RQP are push-based meaning data processing is started by scanning the query result and pushing each tuple down to the leaves of the query tree.

  7. Problem Statement Given, • SQL query Q • Schema SD of relational database (including integrity constraints) • Table R Required : Find a database instance D such that: R = Q(D) D is compliant with SD and its integrity constraints

  8. Problem Statement • There are many different database instances which can be generated for a given Q and R. • Depending on the application, some of these instances might be better than others. • For functional testing, RQP should generate a small D that satisfies the correctness criteria ( R=Q(D) ), so that the running time of tests is reduced.

  9. RQP Architecture

  10. RQP Architecture … • Parser • Traditional query tree is translated into a reverse query tree. • In the reverse query tree, each operator of the relational algebra is translated into a corresponding reverse relational algebra operator.

  11. RQP Architecture … 2.Bottom Up Query Annotation • annotates each operator of a reverse query tree with an input schema SIN and an output schema SOUT . 3. Query Optimization • Query optimization for RQP can be much more aggressive Traditional query optimization because it is acceptable to generate a different D for the same input as long as the criterion R = Q(D) is fulfilled. • not important to carry out join reordering because joins in RQP are mostly cheap.

  12. RQP Architecture … 4. Top Down Data Instantiation • a physical implementation for each operator of the reverse relational algebra that is used for reverse query execution. • For parameterized query top-down data instantiation can use the same annotated reverse query tree for each set of parameter settings

  13. RQP Example The database schema of the Lineitem and Orders tables with their integrity constraints SQL query that asks for the sales (SUM(price)) by orderdate.

  14. RQP Example

  15. Reverse Relational Algebra • Each operator of the relational algebra has a corresponding operator in the reverse relational algebra. • the operators of the RRA are marked as op−1 (e.g reverse of  is  −1 ) • the following equation holds for all operators and all valid tables R: op(op−1(R)) = R

  16. Reverse Relational Algebra • Reverse operators in RRA should not be confused with inverse operators because op−1(op(S)) = S is not necessarily true for some valid tables S. pid>50 -1pid>50

  17. Reverse Relational Algebra … • An operator of the RRA has exactly one input and produces 0 or more output relations. Basic Operators: The reverse variants of the basic operators of the (extended) relational algebra form the basis of the RRA. All other operators of the RRA can be expressed as compositions of these basic operators Algebraic Laws: The relational algebra has laws on associativity, commutativity, etc. on many of its operators. Some laws are not applicable for the RRA (e.g., applying projections before joins).

  18. Reverse Projection (  -1) The reverse projection operator (  -1) generates new columns according to its output schema.

  19. Reverse Selection (  -1) • Returns a superset of input. • Error is returned if input does not match the selection predicate. • If additional tuple are generated than they must satisfy the negation of the selection predicate.

  20. Reverse Aggregation ( x-1) • The reverse aggregation operator (x-1) generates columns according to database schema. • The reverse aggregation operator generates additional rows in order to meet all constraints of its aggregate functions • Returns error if not able to ensure x(x-1 (R)) = R

  21. Reverse Join ( ) • It takes one relation as input and generates two output relations. • The reverse join makes sure that its outputs meet the specified output schemas

  22. Reverse Union (U-1) • The reverse union operator (∪−1) takes one relation as input and generates two output relations. • According to the constraints of the output schemas, the reverse union distributes the tuples of the input relation to the corresponding output relations.

  23. Reverse Minus ( -1) • Input tuples are always routed to the left branch or result in an error. • it is possible that the Reverse Minus operator ( − -1 ) generates new tuples for both branches in order to meet all its constraints

  24. RQP Example

  25. Bottom-up Query Annotation • The bottom-up query annotation phase annotates each operator (op−1 ) of a reverse query tree with an output schema SOUT and an input schema SIN. • Each operator can check the correctness of the input and ensure that it generates valid output • Both schemas (input and output) are defined by (1) the attributes A (names and data types) (2) the integrity constraints C, and (3) the functional dependencies F (4) join dependencies J

  26. Bottom-up Query Annotation • Schema of R <a int primary key, p int> • Select a from R where p=3

  27. Top-down Data Instantiation • The Top-down data instantiation component interprets the optimized reverse query execution plan using an RTable R and possibly query parameters as input. • The iterators are push-based. • The whole data instantiation is started by scanning the RTable and pushing each tupleof the RTable one at a time to the children operators • A push-based model is required because operators of the RRA can have multiple outputs

  28. Top-down Data Instantiation All iterators have the same interface which contains the following three methods: • open(): prepare the iterator for producing data as in traditional query processing; • pushNext(Tuple t): (a) receive a tuple t (b) check if t satisfies the input schema SIN of the operator, (c) produce zero or more output tuples, and (d) for each output tuple, call the pushNext method of the relevant children operators; • close(): clean up everything as in traditional query processing.

  29. Model Checker* • Given a model of a system, tests automatically whether this model meets a given specification. • Mathematical formulation of the constraints and the system – Predicate Logic. • Often generate a model that satisfy or does not satisfy a given formula. • Examples • CVC3 • Alloy Not in the paper

  30. CVC3 Example* Input file for cvc3 % Possible Values for person data type DATATYPE PERSON = P1|P2|P3|P4|P5 END; %Possible values for CAR data type DATATYPE CAR= C1|C2|C3|C4|C5 Not in the paper

  31. CVC3 Example* • Query • CHECKSAT R[1].0=P1 AND R[1].1=C1; • Response • Unsatisfiable • Query • Query R[1].0=P1 AND R[1].1=C1; • Countermodel; • Response • ASSERT (R[1]=(P2,C1)); • ASSERT (R[1]=(P1,C2)); • ASSERT (R[1]=(P3,C3)); • ASSERT (R[1]=(P4,C4)); • ASSERT (R[1]=(P5,C5)); Not in the paper

  32. Top-down Data Instantiation • SPQR is a RQP prototype for functional testing. • The physical algebra of SPQR tries to keep the generated database as small as possible. • In order to generate values for new columns, the reverse operators calls the decision procedure of a model checker • The model checker is treated as a black box. It takes a constraint formula as input and returns one of the possible data instantiations on all variables as output

  33. SQPR example • Example: • Query: select A from R where A + B < 30 • Consider the reverse projection operator. • Input schema • Call Model Checker with the formula A=3 & A+B<30 • Instantiated data

  34. Reverse Projection in SPQR

  35. Instantiate Data in SQPR

  36. An example • For the tuple of the RTable (SUM(price) = 120), thefollowing formula is generated for n = 1,where n is number of tuples to be generated. sum_price=120 & avg_price<=100 & sum_price=price1 & avg_price=sum_price/1 • This formula is given to the decision procedure of the model checker. The model checker cannot find values for the variables price1 and avg price that meet all constraints.

  37. Example contd.. • In the second attempt for n = 2, the following formula is passed to the decision procedure: sum_price=120 & avg_price<=100 & sum_price=price1+price2 & avg_price=sum price/2 • The decision procedure can now find an instantiation: sum_price=120, avg_price=60, price1=80, price2=40,

  38. Reverse Aggregation in SPQR

  39. Processing Nested Queries for SQPR • SPQR uses the concept of nested iterations in a reverse way • The inner subquery can be thought of as a reverse query tree whose input is parameterized on values generated for correlation variables of the outer query • Reverse processing of nested queries is expensive having quadratic complexity with the size of the RTable

  40. Optimization of Data Instantiation • Reverse query processing heavily relies on calls to a model checker. These calls are expensive. In the worst case, the cost is exponential to the size of the formula. • Independent attribute: An attribute a is independent with regard to an output schema SOUT of an operator iff SOUT has no integrity constraints limiting the domain of a and a is not correlated with another attribute a′ (e.g. by a> a′ ) which is not independent. For e.g. SOUT (A,B,C) A=3 & A+B < 20 then C can be considered as a independent attribute

  41. Optimization of Data Instantiation • Constrictiveindependent attribute: An attribute a is constrictive independent, if it is independent with regard to an output schema SOUTdisregarding certain optimization dependent integrity constraints. For e.g. SOUT (A,B,C) A=3 & A+B < 20 & C is unique, then C can be considered as a constrictive independent attribute.

  42. Optimization of Data Instantiation • Default-value Optimization: Assignsa default (fixed) value to an independent attribute a depending on the type of the attribute. • Unique-value Optimization: Assigns a unique increment counter value to a constrictive independent attribute a, which is only bound by unique or primary key constraints. Attributes which use this optimization are not included in the constraint formula.

  43. Optimization of Data Instantiation • Single-value Optimization: Can be applied to a constrictive independent attribute a which is only bound by CHECK constraints. Only included in a constraint formula, the first time the top-down phase needs to instantiate a value for them. The instantiated value is then reused.

  44. Optimization of Data Instantiation • Aggregation-value Optimization: Can be applied to constrictive independent attributes a which are involved in an aggregation. • If SUM(a:float) is an attribute in the operator’s input schema, MIN(a) and MAX(a) are not in the operator’s input schema. Instantiate a value for a by solving a=SUM(a)/n. • If MIN(a) or MAX(a) are in the operator’s input schema, and n ≥ 3. Use values for MIN(a) or MAX(a) once to instantiate a. Instantiate the other values for a by solving a=(SUM(a)-MIN(a)-MAX(a))/(n-2). • a is of data type integer. We can directly compute a by solving SUM(a)=n1×a1+n2 × a2, where a1=⌊sum(a)/n⌋, a2=⌈sum(a)/n⌉,n1=n − n2 and n2=(SUM(a) modulo n). • If only COUNT(a) is in the operator’s input schema,a can be set using the default-value optimization.

  45. Optimization of Data Instantiation • Count heuristics: Does not find instantiations. But reduces the number of attempts for guessing the number of tuples to reverse process an aggregation by constraining the value of n. • Heuristics used are: • If SUM(a) and AVG(a) are attributes of the operator’s input schema, then n=SUM(a)/AVG(a). • If SUM(a) and MAX(a) are attributes of the operator’s input schema, then n ≥ SUM(a)/MAX(a) (if SUM(a) and MAX(a) ≥ 0; if SUM(a) and MAX(a) ≤ 0 use n ≤ SUM(a)/MAX(a)). • If SUM(a) and MIN(a) are attributes of the operator’s input schema, then n ≤ SUM(a)/MIN(a) (if SUM(a) and MIN(a) ≥ 0; if SUM(a) and MIN(a) ≤ 0 use n ≥ SUM(a)/MIN(a)).

  46. Optimization of Data Instantiation • Tolerance on precision: Tolerances can be exploited in order to speed up model checking. Rather than, say, specifying a= 100, a more flexible constraint 90 ≤ a ≤ 110 can be used. Only legal for certain applications. Set to 0 percent by default. • Memoization: Cache calls to the model checker. Useful for reverse operator that often solve similar constraints and carry out the same kind of guessing. The results of guessing for the  −1 operator can be re-used by the x−1 operator

  47. Performance Experiments and Results • The SPQR system was implemented in Java 1.4 installed on Linux AMD Opteron 2.2 GHz Server 4 GB of main memory • As a backend database system PostgreSQL 7.4.8 was used. • Cogent as a decision procedure was used. • SPQR was configured to allow 0 percent tolerance

  48. Performance Experiments and Results • Table 1 shows the size of the databases generated by SPQR for all queries on the three scaling factors ( 100M , 1G, 10G).

  49. Performance Experiments and Results

  50. Performance Experiments and Results • Queries which include an explicit or implicit COUNT value in R, the size of the generated database depends on that COUNT value. • For those queries which do not define a COUNT value, only a small number of tuples are generated

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